How can I composite Eevee and Cycles together in one animation sequence?
Organize your outliner like this:

Add a scene and select Linked Copy.

I'm calling mine Eevee.
Set up Eevee scene's outliner like this:

In the Eevee scene, set up the shaders of the objects in the Cycles collection like so:

Be sure to set the Blend Mode to Alpha Clip (unless otherwise necessary).

Back in the Cycles scene, set up the compositing something like this:

And be sure to enable transparency in the film settings:

When you press F12, both scenes will render automatically.